Expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta and Gamma functions

Considering the equations for some functions involving the first or the second derivatives of the biconfluent Heun function, we construct two expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta functions. The first series applies single Beta functions as expansion functions, while the second one involves a combination of two Beta functions. The coefficients of expansions obey four- and five-term recurrence relations, respectively. It is shown that the proposed technique is potent to produce series solutions in terms of other special functions. Two examples of such expansions in terms of the incomplete Gamma functions are presented.     

An examination of the steady-state solution of the differential equation governing indoor sound fields

The use of the characteristic functions to obtain a steady-state sound field is based on the completeness of the set of the characteristic functions. For absorptive boundaries, the characteristic functions are complex functions, and there is no proof in mathematics of whether the set of the characteristic functions is complete. In addition, the use of the characteristic functions to obtain a steady-state sound field of three-dimensional space requires the solving of a huge amount of the complex number k_nx The aims of this paper are to examine whether the set of the characteristic functions can still be complete for strong absorptive boundaries and to provide a practical solving method for k_nx.     

Current Algebra on the Torus

We derive the N-point one-loop correlation functions for the currents of an arbitrary affine Kac-Moody algebra. The one-loop amplitudes, which are elliptic functions defined on the torus Riemann surface, are specified by group invariant tensors and certain constant tau-dependent functions. We compute the elliptic functions via a generating function, and explicitly construct the invariant tensor functions recursively in terms of Young tableaux. The lowest tensors are related to the character formula of the representation of the affine algebra. These general current algebra loop amplitudes provide a building block for open twistor string theory, among other applications.     

Correlated basis functions for the many-electron problem

The method of correlated basis functions is applied to the electron system in a metal. To overcome the twofold difficulty of large particle number and long-range Coulomb interaction in metals, a new optimal cluster decomposition for arbitrary correlated wave functions is derived. With this method, not only ground-state properties, but also thermal averages and response functions can be calculated from a given set of correlated basis functions. The appropriate synthesis of correlated wave functions including physically expected properties of subsystems as for instance partially filled inner shells in transition metals is discussed in detail.     

On the validity of the inclusion of geometrical shadowing functions in the multiple-scatter Kirchhoff approximation

The validity of the geometrical shadowing functions for use in calculations of the light distribution scattered from a rough surface is investigated. By the use of the multiple-scatter Kirchhoff approximation, single- and double-scattered contributions are calculated with and without the incident, intrasurface and scatter-shadowing functions. Explicit conditions are given for the validity of the geometrical shadowing functions.     

Some extensions of the Fermi-Dirac and Bose-Einstein functions with applications to the family of the zeta and related functions

The familiar Fermi-Dirac and Bose-Einstein functions are of importance not only for their rôle in quantum statistics, but also for their several interesting mathematical properties in themselves. Here, in our present investigation, we have extended these functions by introducing an extra parameter in a way that gives new insights into these functions and their relationship to the family of zeta functions. These extensions are dual to each other in a sense that is explained in this paper. Some identities are proved here for each of these general functions and their relationship with the general Hurwitz-Lerch zeta function Φ(z, s, a) is exploited to derive some other (presumably new) identities.     

On the structure of the conformal covariant N-point functions

The conformal covariant n-point functions for the fields with arbitrary spin and scale dimension are found. These functions are determined up to a set of arbitrary functions on $\text{n(n?3)}\text{2}$ independent conformal invariant variables. These arbitrary functions cannot be determined without dynamical assumptions.     

On the Embedding of von Neumann Subalgebras

For a von Neumann algebra with a cyclic and separating vector it will be shown that the von Neumann subalgebras with the same cyclic vector can uniquely be characterized by one-parametric operator-valued functions obeying a set of conditions. Since the properties contain no reference to the subalgebra these operator-valued functions will be called characteristic functions. On the set of characteristic functions there exists a natural topology under which this set is complete. Received: 3 December 1998 /Accepted: 15 February 1999     

Connection between the 2-body energy of the Kaxiras-Pandey and the Biswas-Hamann potentials

Relationship among interatomic potential functions can be useful in shedding insight on the extent of similarity, and in obtaining a potential function from parameters of another potential function. The 2-body portion of the Biswas-Hamann (BH) and the Kaxiras-Pandey (KP) potential functions are related by equating both functions, as well as their corresponding derivatives up to the third order at the equilibrium bond length. Validity of the parametric relationship is verified by plotting the loose form of the 2-body BH potential in terms of KP parameters and comparing it with the KP potential function. The parametric relationships developed herein are then compared with those that concern other potential functions, with particular emphasis on the scaling factors.     

Comparing the evolution of canalyzing and threshold networks

We study the influence of the type of update functions on the evolution of Boolean networks under selection for dynamical robustness. The chosen types of functions are canalyzing functions and threshold functions. Starting from a random initial network, we evolve the network by an adaptive walk. During the first time period, where the networks evolve to the plateau of 100 percent fitness, we find that both type of update functions give the same behavior, albeit for different network sizes and connectedness. However, on the long run, as the networks continue to evolve on the fitness plateau, the different types of update functions give rise to different network structure, due to their different mutational robustness. When both types of update functions occur together, none of them is preferred under long-term evolution.     

Lie group theory of the bessel equation of the first kind of integral order

We discuss the Bessel differential equation of the first kind of integral order and the associated functions from a Lie-group-theoretical background. All the familiar properties of the Bessel equation and functions are obtained. The analytic methodology developed in the study can easily be adapted to the study of some other special functions of mathematical physics.     

Nuclear shape measurement and the validity of the collective model for light nuclei

We have carried out calculations of collective transition properties using wave functions projected from a range of HF wave functions of a wide range of basis dimensionalities and corresponding to different effective interactions. Particular calculations show that the best HF wave functions have multipole moments which are closer to experimental values than elementary analysis has suggested. More general results, intended to be useful in the interpretation of inelastic scattering, are given.     

Plasma response functions,fluctuation-dissipation relations and the velocity-average-approximation

We derive and list relationships for quadratic and cubic static response functions and connect them with three- and four-point functions through generalized fluctuation-dissipation relations. We also introduce the useful concept of “response function of the second kind.” to describe the reponse of a system through the perturbation of its two-point functions. Next, we point out that the VAA (velocity-average-approximation) introduced earlier as a part of a dynamical approximation scheme for strongly coupled one-component plasmas is exact in the static limit. This observation, combined with the fluctuation-dissipation relations, allows one to derive a hierarchy of equations for response functions of increasing nonlinearity, which is equivalent to the BGY hierarchy for particle distribution functions.     

Rate-intensity functions in the emu auditory nerve

Rate-versus-intensity functions recorded from mammalian auditory-nerve fibers have been shown to form a continuum of shapes, ranging from saturating to straight and correlating well with spontaneous rate and sensitivity. These variations are believed to be a consequence of the interaction between the sensitivity of the hair-cell afferent synapse and the nonlinear, compressive growth of the cochlear amplifier that enhances mechanical vibrations on the basilar membrane. Little is known, however, about the cochlear amplifier in other vertebrate species. Rate-intensity functions were recorded from auditory-nerve fibers in chicks of the emu, a member of the Ratites, a primitive group of flightless birds that have poorly differentiated short and tall hair cells. Recorded data were found to be well fitted by analytical functions which have previously been shown to represent well the shapes of rate-intensity functions in guinea pigs. At the fibers' most sensitive frequencies, rate-intensity functions were almost exclusively of the sloping (80.9%) or straight (18.6%) type. Flat-saturating functions, the most common type in the mammal, represented only about 0.5% of the total in the emu. Below the best frequency of each fiber, the rate-intensity functions tended more towards the flat-saturating type, as is the case in mammals; a similar but weaker trend was seen above best frequency in most fibers, with only a small proportion (18%) showing the reverse trend. The emu rate-intensity functions were accepted as supporting previous evidence for the existence of a cochlear amplifier in birds, the conclusion was drawn further that the nonlinearity observed is probably due to saturation of the hair-cell transduction mechanism.     

The scaling limit and Osterwalder-Schrader axioms for the two-dimensional Ising model

From a Feynman-Kac formula in a Fermion Fock space for the Schwinger functions of the infinite lattice periodic two-dimensional Ising model, scaled and scaling limit Schwinger functions are defined and shown to admit an absolutely convergent series representation. As the critical temperature is attained, it is shown that the scaled Schwinger functions converge and that the resulting scaling limit Schwinger functions obey the Osterwalder-Schrader axioms.     

Theory of the Ionization Equilibrium in Nonideal Alkali Plasmas

With the help of local pseudopotentials and binary distribution functions bound and scattering state parts of the thermodynamic functions of nondegenerate nonideal alkali plasmas with a large interaction parameter are calculated. The one-particle distribution functions for the atomic levels are obtained by reinterpretation from the bound state part of the binary distribution functions. The ionization equilibrium is described by a Saha equation which is evaluated for certain ranges of temperature and density. The transition to high conducting states is discussed.     

The behavior of the weyl function in the zero-dispersion KdV limit

The moment formulas that globally characterize the zero-dispersion limit of the Korteweg-deVries (KdV) equation are known to be expressed in terms of the solution of a maximization problem. Here we establish a direct relation between this maximizer and the zero-dispersion limit of the logarithm of the Jost functions associated with the inverse spectral transform. All the KdV conserved densities are encoded in the spatial derivative of these functions, known as Weyl functions. We show the Weyl functions are densities of measures that converge in the weak sense to a limiting measure. This limiting measure encodes all of the weak limits of the KdV conserved densities. Moreover, we establish the weak limit of spectral measures associated with the Dirichlet problem. Dedicated to Peter Lax on his 70^th birthday     

Correlation function of the amplitude and of the intensity fluctuation for a laser model near threshold

In extension of a preceding paper the correlation function of the amplitude and of the intensity fluctuation are calculated in the threshold region. The laser amplitude is treated as a classical random variable obeying a van der Pol equation with a noise term. In order to get correlation functions, the method of distribution functions is employed. The distribution functions are evaluated by the Fokker-Planck equation. The lowest eigensolutions of the Fokker-Planck equation are obtained approximately by a variational method.     

Hyperspherical harmonics and the nuclear four-body problem

A method proposed earlier by Aguilera, Moshinsky, and Kramer, for adapting a system of translationally invariant four-particle harmonic oscillator functions to the symmetry of the permutation group S(4), is applied to hyperspherical harmonic functions depending on three relative vectors. Except for a few cases in which diagonalization of matrices is required, the method gives closed formulas for orthonormal sets of harmonic functions with good permutational symmetry. The matrix elements of S(4) permutations with respect to the harmonic functions are obtained.     

A unified formulation of the coherent apodisation problem

A unified formulation of the coherent apodisation problem is presented as a generalised variance problem where the concentrations of the functions in the spatial and frequency domains are measured in terms of arbitrary weighted energy distributions. The solution of this optimisation results in solving a Fredholm equation, with different pairs of weighting functions, to obtain various apodisation functions.